Properties

Label 43.4
Modulus $43$
Conductor $43$
Order $7$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(43, base_ring=CyclotomicField(14))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4]))
 
pari: [g,chi] = znchar(Mod(4,43))
 

Basic properties

Modulus: \(43\)
Conductor: \(43\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(7\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 43.e

\(\chi_{43}(4,\cdot)\) \(\chi_{43}(11,\cdot)\) \(\chi_{43}(16,\cdot)\) \(\chi_{43}(21,\cdot)\) \(\chi_{43}(35,\cdot)\) \(\chi_{43}(41,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{7})\)
Fixed field: 7.7.6321363049.1

Values on generators

\(3\) → \(e\left(\frac{2}{7}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(1\)\(1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{4}{7}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 43 }(4,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{43}(4,\cdot)) = \sum_{r\in \Z/43\Z} \chi_{43}(4,r) e\left(\frac{2r}{43}\right) = -2.407313842+6.09957704i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 43 }(4,·),\chi_{ 43 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{43}(4,\cdot),\chi_{43}(1,\cdot)) = \sum_{r\in \Z/43\Z} \chi_{43}(4,r) \chi_{43}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 43 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{43}(4,·)) = \sum_{r \in \Z/43\Z} \chi_{43}(4,r) e\left(\frac{1 r + 2 r^{-1}}{43}\right) = 7.9466717597+-9.9648112031i \)